Integrand size = 17, antiderivative size = 117 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {2 c^2 \left (1+a^2 x^2\right )}{15 a}-\frac {c^2 \left (1+a^2 x^2\right )^2}{20 a}+\frac {8}{15} c^2 x \arctan (a x)+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {4 c^2 \log \left (1+a^2 x^2\right )}{15 a} \]
-2/15*c^2*(a^2*x^2+1)/a-1/20*c^2*(a^2*x^2+1)^2/a+8/15*c^2*x*arctan(a*x)+4/ 15*c^2*x*(a^2*x^2+1)*arctan(a*x)+1/5*c^2*x*(a^2*x^2+1)^2*arctan(a*x)-4/15* c^2*ln(a^2*x^2+1)/a
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.56 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\frac {c^2 \left (-14 a^2 x^2-3 a^4 x^4+4 a x \left (15+10 a^2 x^2+3 a^4 x^4\right ) \arctan (a x)-16 \log \left (1+a^2 x^2\right )\right )}{60 a} \]
(c^2*(-14*a^2*x^2 - 3*a^4*x^4 + 4*a*x*(15 + 10*a^2*x^2 + 3*a^4*x^4)*ArcTan [a*x] - 16*Log[1 + a^2*x^2]))/(60*a)
Time = 0.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5413, 27, 5413, 5345, 240}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \arctan (a x) \left (a^2 c x^2+c\right )^2 \, dx\) |
\(\Big \downarrow \) 5413 |
\(\displaystyle \frac {4}{5} c \int c \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {c^2 \left (a^2 x^2+1\right )^2}{20 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{5} c^2 \int \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {c^2 \left (a^2 x^2+1\right )^2}{20 a}\) |
\(\Big \downarrow \) 5413 |
\(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {c^2 \left (a^2 x^2+1\right )^2}{20 a}\) |
\(\Big \downarrow \) 5345 |
\(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {c^2 \left (a^2 x^2+1\right )^2}{20 a}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {c^2 \left (a^2 x^2+1\right )^2}{20 a}\) |
-1/20*(c^2*(1 + a^2*x^2)^2)/a + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x])/5 + (4 *c^2*(-1/6*(1 + a^2*x^2)/a + (x*(1 + a^2*x^2)*ArcTan[a*x])/3 + (2*(x*ArcTa n[a*x] - Log[1 + a^2*x^2]/(2*a)))/3))/5
3.2.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65
method | result | size |
parts | \(\frac {c^{2} \arctan \left (a x \right ) a^{4} x^{5}}{5}+\frac {2 c^{2} \arctan \left (a x \right ) a^{2} x^{3}}{3}+c^{2} x \arctan \left (a x \right )-\frac {c^{2} a \left (\frac {3 a^{2} x^{4}}{4}+\frac {7 x^{2}}{2}+\frac {4 \ln \left (a^{2} x^{2}+1\right )}{a^{2}}\right )}{15}\) | \(76\) |
derivativedivides | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {2 a^{3} c^{2} x^{3} \arctan \left (a x \right )}{3}+a \,c^{2} x \arctan \left (a x \right )-\frac {c^{2} \left (\frac {3 a^{4} x^{4}}{4}+\frac {7 a^{2} x^{2}}{2}+4 \ln \left (a^{2} x^{2}+1\right )\right )}{15}}{a}\) | \(80\) |
default | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {2 a^{3} c^{2} x^{3} \arctan \left (a x \right )}{3}+a \,c^{2} x \arctan \left (a x \right )-\frac {c^{2} \left (\frac {3 a^{4} x^{4}}{4}+\frac {7 a^{2} x^{2}}{2}+4 \ln \left (a^{2} x^{2}+1\right )\right )}{15}}{a}\) | \(80\) |
parallelrisch | \(-\frac {-12 c^{2} \arctan \left (a x \right ) a^{5} x^{5}+3 a^{4} c^{2} x^{4}-40 a^{3} c^{2} x^{3} \arctan \left (a x \right )+14 a^{2} c^{2} x^{2}-60 a \,c^{2} x \arctan \left (a x \right )+16 c^{2} \ln \left (a^{2} x^{2}+1\right )}{60 a}\) | \(85\) |
risch | \(-\frac {i c^{2} x \left (3 a^{4} x^{4}+10 a^{2} x^{2}+15\right ) \ln \left (i a x +1\right )}{30}+\frac {i c^{2} a^{4} x^{5} \ln \left (-i a x +1\right )}{10}-\frac {c^{2} a^{3} x^{4}}{20}+\frac {i c^{2} a^{2} x^{3} \ln \left (-i a x +1\right )}{3}-\frac {7 c^{2} a \,x^{2}}{30}+\frac {i c^{2} x \ln \left (-i a x +1\right )}{2}-\frac {4 c^{2} \ln \left (-a^{2} x^{2}-1\right )}{15 a}-\frac {49 c^{2}}{180 a}\) | \(137\) |
meijerg | \(\frac {c^{2} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{4 a}+\frac {c^{2} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2 a}+\frac {c^{2} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4 a}\) | \(172\) |
1/5*c^2*arctan(a*x)*a^4*x^5+2/3*c^2*arctan(a*x)*a^2*x^3+c^2*x*arctan(a*x)- 1/15*c^2*a*(3/4*a^2*x^4+7/2*x^2+4/a^2*ln(a^2*x^2+1))
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {3 \, a^{4} c^{2} x^{4} + 14 \, a^{2} c^{2} x^{2} + 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, {\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \arctan \left (a x\right )}{60 \, a} \]
-1/60*(3*a^4*c^2*x^4 + 14*a^2*c^2*x^2 + 16*c^2*log(a^2*x^2 + 1) - 4*(3*a^5 *c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x)*arctan(a*x))/a
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {a^{3} c^{2} x^{4}}{20} + \frac {2 a^{2} c^{2} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {7 a c^{2} x^{2}}{30} + c^{2} x \operatorname {atan}{\left (a x \right )} - \frac {4 c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{15 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**4*c**2*x**5*atan(a*x)/5 - a**3*c**2*x**4/20 + 2*a**2*c**2*x* *3*atan(a*x)/3 - 7*a*c**2*x**2/30 + c**2*x*atan(a*x) - 4*c**2*log(x**2 + a **(-2))/(15*a), Ne(a, 0)), (0, True))
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {1}{60} \, {\left (3 \, a^{2} c^{2} x^{4} + 14 \, c^{2} x^{2} + \frac {16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \arctan \left (a x\right ) \]
-1/60*(3*a^2*c^2*x^4 + 14*c^2*x^2 + 16*c^2*log(a^2*x^2 + 1)/a^2)*a + 1/15* (3*a^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*arctan(a*x)
\[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right ) \,d x } \]
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.59 \[ \int \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {c^2\,\left (16\,\ln \left (a^2\,x^2+1\right )+14\,a^2\,x^2+3\,a^4\,x^4-40\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )-12\,a^5\,x^5\,\mathrm {atan}\left (a\,x\right )-60\,a\,x\,\mathrm {atan}\left (a\,x\right )\right )}{60\,a} \]